ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 4 Issue 2 (2012.), Pages 83-90
ON BASES IN THE SPACE OF VECTOR VALUED ENTIRE DIRICHLET SERIES OF TWO COMPLEX VARIABLES
(COMMUNICATED BY INDRAJIT LAHIRI)
G. S. SRIVASTAVA AND ARCHNA SHARMA
Abstract. The space of all entire functions represented by vector valued Dirichlet series of two complex variables is considered in this paper. It is equipped with two equivalent topologies. The main result of this paper is con-cerned with finding the conditions for a base inXto become a proper base and certain continuous linear operators which are used to determine the proper bases inX.
1. Introduction
Consider
f(s1, s2) = ∞ X
m,n=0
am,nexp(λms1+µns2), (sj=σj+itj, j= 1,2) (1)
wherea0m,nsbelong to a commutative Banach algebra (E,||.||);0 =λ0< λ1< ... < λm→ ∞asm→ ∞,0 =µ0< µ1< ... < µn→ ∞as n→ ∞ and
lim m+n→∞sup
log(m+n)
λm+µn
=D <+∞, (2)
lim m+n→∞sup
log(||am,n||)
λm+µn
=−∞ . (3)
Then the Dirichlet series given in (1) represents a vector valued entire function
f(s1, s2) in two complex variables (see[3]). LetXdenote the space of all entire functions f :C2→Edefined as above by the vector valued Dirichlet series (1). In [1] and [2], S.Daoud obtained the properties of space of entire functions defined by Dirichlet
series of two complex variables. In this paper we study the properties of the space
X of all entire functions defined by vector valued Dirichlet series.
2000Mathematics Subject Classification. 32A15, 30H05.
Key words and phrases. Vector Valued Dirichlet series, Entire functions, Linear operator. c
2012 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.
Submitted February 04, 2012. Revised Version Submitted on: April 06, 2012, Accepted April 30,2012.
2. Different Topologies and Bases in X
Let us assume that{σ1(k)}and{σ2(k)} are two non-decreasing sequences of positive numbers, σ1(k) → ∞and σ2(k) → ∞ with k → ∞.For each f ∈ Xgiven by (1), we define
||f;σ1(k), σ(2k)|| = ∞ X
m,n=0
||am,n||exp(λmσ
(k) 1 +µnσ
(k)
2 ). (4)
Now from (3) we obtain
lim
m+n→∞inf log||am,n||
−1/ (λm+µn)= +∞.
Then for an arbitrarily large numberK ,we have form+n > N0,
||am,n|| < exp[−K(λm+µn)].
Hence ||f;σ1(k), σ2(k)|| = ∞ X
m,n=0
||am,n||exp(λmσ
(k) 1 +µnσ
(k) 2 )
= N0 X
m+n=0
||am,n||exp(λmσ
(k) 1 +µnσ
(k) 2 ) +
∞ X
m+n≥N0
||am,n||exp(λmσ
(k) 1 +µnσ
(k) 2 )
< O(1) +P∞
m+n≥N0exp[λm(σ
(k)
1 −K) +µn(σ
(k) 2 −K)]
< ∞,sinceK >> σ1(k), σ(2k)can be chosen.
Then the series on right hand side of (4) converges and ||f;σ1(k), σ(2k)|| defines a norm onX for eachk = 1,2...Further, from (4), it follows that
||f;σ1(k), σ(2k)|| ≤ ||f;σ1(k+1), σ(2k+1)||, for all k≥1.
With these countable number of norms||f;σ1(k), σ2(k)||,(k≥1), we define a metric topology onXwith metricρdefined as
ρ(f, g) =X k≥1
1 2k.
||f−g;σ(1k), σ2(k)||
1 +||f−g;σ1(k), σ2(k)|| , f, g∈ X.
For eachf ∈ X and0< σ1, σ2<∞, put
M(f;σ1, σ2) = sup −∞<t1,t2<∞
||f(σ1+it1, σ2+it2)|| (5)
Then M(f, σ1, σ2) defines a family of norms on X. Using this we define a metric
topology generated by norms (5) as
=(f, g) = ∞ X
j=1
1 2j
M(f−g, σ(1j), σ2(j))
1 +M(f−g, σ(1j), σ2(j))
where {σ1(j)} and {σ2(j)} are two non decreasing sequences of positive numbers,
σ1(j)→ ∞andσ2(j)→ ∞asj→ ∞. It was proved earlier in [1] that forα >0
M(f;σ1, σ2)≤ ||f;σ1, σ2|| ≤K(α)M(f;σ1+α, σ2+α)
whereK = P∞
m,n=0exp[−α(λm+µn)].Hence the two topologies defined onX byρand =are equivalent.
Now we give the characterizations of certain types of bases inX.
A sequence{fm,n: m, n≥0} ⊂ X is said to be a base forX if for each f ∈ X, there exists a unique sequence{Cm,n:m, n≥0} ⊂E,such thatf =P∞m,n=0Cm,nfm,n where the convergence of the infinite series being with respect to the topology on
X.
Here{Cm,n}are called the base functions. Ifem,n∈X, em,n(s1, s2) = exp(s1λm+
µns2), m, n ≥ 1then each f ∈ Xcan be expressed as in (1) with coefficients
{am,n}satisfying
lim m+n→∞sup
log(||am,n||)
λm+µn
=−∞ .
Therefore{em,n}is a base inX .
A base{fm,n}will be called a genuine base forX if the corresponding base functions satisfy (3).A sequence{fm,n}will be called an absolute base forX if it is a base in
Xand the infinite series corresponding to each f ∈ Xis absolutely convergent with respect to the topology on X.A sequence{fm,n}will be called a proper base forX if it is a genuine and an absolute base forX.
3. Main Results
We now give the characterization of proper bases. We prove
Theorem 3.1. Let {Cm,n} be an arbitrary sequence contained inE satisfying
lim m+n→∞sup
log(||Cm,n||)
λm+µn
=−∞. (6)
and {αm,n : m, n ≥ 0} ⊂ X, then the series ΣM(Cm,nαm,n;σ1,σ2) converges if
and only if
lim m+n→∞
logM(αm,n;σ1, σ2)
λm+µn
<∞. (7)
for eachσ1, σ2.
Proof. Let us assume that equation (7) is satisfied. Then for
σ1, σ2>0 there exists a constantε1=ε1(σ1,σ2)>0 such that
logM(αm,n;σ1, σ2)< ε1(λm+µn) (8) form+n≥N(ε1).
Letε2> ε1, then using (6) we find that existsN1=N1(ε2),such that
||Cm,n|| ≤ exp{−ε2(λm+µn)} ; m+n≥N1. (9)
From (8) and (9),we get
∀m+n≥N2=max(N, N1). In an earlier paper, we proved the following result in [?] :
µ(f, σ1, σ2)≤M(f, σ1, σ2)≤Kµ(f, σ1+α, σ2+α) whereK=K(α)
andµ(f, σ1, σ2) = max
m,n≥0{||am,n||exp(λmσ1+µnσ2)}denotes the maximum term
of the entire functionf(s1, s2). Using the above result, we find that
P∞
m+n=0M(Cm,nαm,n;σ1,σ2) converges for everyσ1, σ2.
Conversely, suppose that (7) is false. Hence for some σ1, σ2 > 0, there exist se-quences{mp}and{nq} such that
logM(αmp,nq;σ1, σ2)>(p+q)(λmp+µnq), p, q≥0. (10)
Now we define the sequence{Cm,n} ⊂E such that
log||Cm,n||=
λm+µn−logM(αm,n;σ1, σ2) f or m=mp;n=nq
−∞ f or m6=mp;n6=nq
Then from (10), we have lim m+n→∞sup
log(||Cm,n||)
λm+µn = −∞.Hence for the sequence
defined above, equation (6) holds but
M(Cm,nαm,n;σ1, σ2) =
Cmp,nqM(αmp,nq;σ1, σ2) = exp(λmp+µnq), m=mp, n=nq,
0 form6=mp , n6=nq
and so ΣM(Cm,nαm,n;σ1,σ2) does not converge. Hence equation (7) is true. This
completes the proof of Theorem 1.
Remark 1. Theorem 3.1 above generalizes Theorem 3.2 of [1]. Next we prove
Theorem 3.2. Let{Cm,n} be an arbitrary sequence contained in E and {αm,n :
m, n≥0} ⊂Xsuch that the series ΣM(Cm,nαm,n;σ1,σ2)converges. Then
lim m+n→∞sup
log(||Cm,n||)
λm+µn
=−∞. (11)
if and only if
lim σ1,σ2→∞
lim m+n→∞inf
logM(αm,n;σ1, σ2) λm+µn
= +∞. (12)
Proof. Let us assume that the equation (12) holds and (11) does not hold. Therefore for eachσ1, σ2>0, series ΣM(Cm,nαm,n;σ1,σ2)converges but
lim m+n→∞sup
log(||Cm,n||)
λm+µn
6
=−∞.
Now there exist sequences{mp},{nq} of positive integers such that
log(||Cm,n||)> α(λmp+µnq);α >−∞
By (12) we can findσ1, σ2, such that
lim m+n→∞inf
logM(αm,n;σ1, σ2)
λm+µn
and logM(Cmp,nqλ αmp,nq;σ1,σ2)
mp+µnq >1, p, q≥0 .
From this we find that ΣM(Cm,nαm,n;σ1,σ2) does not converge .This proves the first part of the result .Conversely, let us take (11) to be true but (12) be not true. Then for someβ >0 and for eachσ1, σ2>0
lim σ1,σ2→∞
lim m+n→∞inf
logM(αm,n;σ1, σ2)
λm+µn
< β <+∞.
SinceM(αm,n;σ1, σ2) is monotonically increasing inσ1, σ2 >0 for each fixed pair of integers(m, n) then there exist sequences{mp},{nq}such that
logM(αmp,nq;σ1, σ2)< β(λmp+µnq).
Now we define{Cm,n} ⊂E as follows
log||Cm,n||=
−β(λm+µn) form=mp;n=nq
−∞ form6=mp;n6=nq
Then for givenσ1, σ2
∞ X
m,n=0
||Cm,n||M(αm,n;σ1, σ2)≤ ∞ X
m,n=0
exp[−β(λm+µn)]<+∞
and thereforeP∞
m,n=0M(Cm,nαm,n;σ1,σ2)converges for positive real numbersσ1, σ2. Now from the above, we get,
lim m+n→∞sup
log(||Cm,n||)
λm+µn
=−β
that is, (11) does not hold. Hence (11) implies (12) and this proves the theorem.
Remark 2. Theorem 3.2 above generalizes Lemma 3.3 of [1]. Our next result characterizes proper bases.
Theorem 3.3. Let{αm,n : m, n ≥ 0}be an absolute base inX, then {αm,n} is a
proper base if and only if (7)and(8) hold good.
Proof. Earlier we have shown that if{αm,n:m, n≥0} is a proper base then
lim m+n→∞sup
logM(αm,n;σ1, σ2)
λm+µn
<∞.
for eachσ1, σ2.The result now follows on using the definitions of genuine and proper bases and the results in Theorems 3.1and 3.2.
Now we obtain the characterizations of linear continuous operators .We prove
Theorem 3.4. Let {αm,n: m, n≥0} ⊂ X.Suppose T is a linear operator from
XintoX, such that T(δm,n) =αm,n m, n≥0 where δm,n(s1, s2) = exp(λms1+
µns2).ThenT is a continuous operator on Xif for eachσ1, σ2≥0
lim m+n→∞sup
logM(αm,n;σ1, σ2) λm+µn
<∞. (13)
Conversely, if equation (13) holds, then there exists a continuous linear operator
T :X →X such that
Proof. Suppose that T is a continuous linear operator from X into X with
T(δm,n) =αm,n.Then for given σ1, σ2> 0, there exists a positive constant K and numbersσ01>0, σ20 >0 such that
M(T δm,n;σ1, σ2) =M(αm,n;σ1, σ2)≤KM(δm,n;σ
0
1, σ
0
2)
≤Kexp{(λm+µn)σ} , σ= max(σ
0
1, σ
0
2).
Hence
lim m+n→∞sup
logM(αm,n;σ1, σ2)
λm+µn
<∞.
Thus equation (13) follows.
Conversely, assume that equation (13) is true. Letα∈X, thenαis represented by
α= ∞ X
m,n=0
am,nδm,n
where the coefficientsam,n’s satisfy equation (3).Since equation (13) holds, therefore there exists anA0=A0(σ1, σ2),depending onσ1, σ2 such that
lim m+n→∞
logM(αm,n;σ1,σ2)
λm+µn ≤A0for allm+n≥N
or
M(αm,n;σ1, σ2)≤expA0(λm+µn) for allm+n≥N
Therefore, noticing that equation (3) is already valid for the coefficientsam,n’s , we find thatPa
m,nδm,nconverges inY and so it represents an elements of X.Hence there is natural transformationT :X→X such that:
T(α) = ∞ X
m,n=0
am,nδm,n
with
α= ∞ X
m,n=0
am,nδm,n.
Clearly T(δm,n) = αm,n , m, n ≥ 0. We are now required to show that Tis continuous onXwith respect to the toplogyT.It is sufficient to show thatT is con-tinuous on (X, ρ).The norms||..., σ1, σ2||are continuous onX , therefore given any two numbersσ1, σ2>0, we find
||T(α), σ1, σ2|| = || lim p+q→∞
p+q X
m,n=0
am,n αm,n, σ1, σ2||
≤ lim
p+q→∞
p+q X
m,n=0
||am,n|| ||αm,n;σ1, σ2||
≤ lim
p+q→∞
p+q X
m,n=0
||am,n|| exp(λm+µn)σ=||α;σ, σ||,
where σ=σ(σ1, σ2).Thus T : (X,||..., σ, σ||)→ (X,||..., σ1, σ2||) is continuous and
Lastly we prove
Theorem 3.5. If T is a linear operator fromX into itself, such that T andT−1
are
continuous then {T(δm,n); m, n≥0} is a proper base in the closed subspace T[X]
of X. Conversely if {αm,n : m, n ≥ 0} is a proper base in a closed subspace
Y ofX, then there exists a continuous linear operator T : X → X , such that
T(δm,n) = αm,n.
Proof. Suppose that T is the linear operator mentioned in the hypothesis. Then
T[X] is a closed subspace ofX.
LetT(δm,n) =αm,n, m, n≥0 and letf ∈T[X], then
T−1(f) = ∞ X
m,n=0
am,nδm,n
wheream,n’s satisfy equation (3) . Now
p+q X
m,n=0
am,nδm,n→T−1(f) (14)
inX asp+q→ ∞.
NowT is continuous and linear, and equation (14) implies
f = ∞ X
m,n=0
am,nαm,n (15)
Since equation (13) holds, thenPM(a
m,nαm,n, σ1, σ2) converges for every real and positiveσ1, σ2and the representation off in (15) is unique. SinceT−1is continuous.
We conclude that{αm,n: m, n≥0} is a proper base forT[X].
Conversely: let {αm,n: m, n≥0} be a proper base for a closed subspaceYofX Hence equation (13) holds. Therefore by Theorem 3.4, there exists a continuous linear operatorT onX into itself, such thatT(δm,n) =αm,n, m, n≥0.
Now let f ∈X, f 6= 0, thenf is represented by equation (1) whose coefficients
am,n’s satisfy equation (3). Thus
T(f) = ∞ X
m,n=0
am,nαm,n6= 0
ThereforeT is one-to-one. HenceT is a continuous algebraic isomorphism fromX
intoX.Now apply the well known theorem of Banach ([1]) we find thatT−1exists
and is continuous .Hence proof of the theorem is complete.
Acknowledgements. The authors are grateful to the referee for his valuable com-ments and suggestions.
References
[1] S. Daoud, Bases in the space of entire Dirichlet function of two complex variables, Coll. Math. Vol. XXXV (1984), pp. 35-42.
[2] S. Daoud, Certain operators in the space of entire function represented by Dirichlet series of several complex variables, Le Matemetiche, Vol. XLVII (1992), Fasc. I, pp.3-8.
[4] G.S.Srivastava and Archna Sharma, Spaces of entire functions represented by vector valued Dirichlet series of two complex variables, (to appear).
Department of Mathematics, Indian Institute of Technology Roorkee,Roorkee-247667, country-regionplaceIndia.
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